45 research outputs found

### Exact results for anomalous transport in one dimensional Hamiltonian systems

Anomalous transport in one dimensional translation invariant Hamiltonian
systems with short range interactions, is shown to belong in general to the KPZ
universality class. Exact asymptotic forms for density-density and
current-current time correlation functions and their Fourier transforms are
given in terms of the Pr\"ahofer-Spohn scaling functions, obtained from their
exact solution for the Polynuclear growth model. The exponents of corrections
to scaling are found as well, but not so the coefficients. Mode coupling
theories developed previously are found to be adequate for weakly nonlinear
chains, but in need of corrections for strongly anharmonic interparticle
potentials.Comment: Further corrections to equations have been made. A few comments have
been added, e.g. on the non-applicability to exactly solved model

### The uphill turtle race: on short time nucleation probabilities

The short time behavior of nucleation probabilities is studied by
representing nucleation as diffusion in a potential well with escape over a
barrier. If initially all growing nuclei start at the bottom of the well, the
first nucleation time on average is larger than the inverse nucleation
frequency. Explicit expressions are obtained for the short time probability of
first nucleation. For very short times these become independent of the shape of
the potential well. They agree well with numerical results from an exact
enumeration scheme. For a large number N of growing nuclei the average first
nucleation time scales as 1/\log N in contrast to the long-time nucleation
frequency, which scales as 1/N. For linear potential wells closed form
expressions are obtained for all times.Comment: 8 pages, submitted to J. Stat. Phy

### A Note on the Ruelle Pressure for a Dilute Disordered Sinai Billiard

The topological pressure is evaluated for a dilute random Lorentz gas, in the
approximation that takes into account only uncorrelated collisions between the
moving particle and fixed, hard sphere scatterers. The pressure is obtained
analytically as a function of a temperature-like parameter, beta, and of the
density of scatterers. The effects of correlated collisions on the topological
pressure can be described qualitatively, at least, and they significantly
modify the results obtained by considering only uncorrelated collision
sequences. As a consequence, for large systems, the range of beta-values over
which our expressions for the topological pressure are valid becomes very
small, approaching zero, in most cases, as the inverse of the logarithm of
system size.Comment: 15 pages RevTeX with 2 figures. Final version with some typo's
correcte

### On thermostats and entropy production

The connection between the rate of entropy production and the rate of phase
space contraction for thermostatted systems in nonequilibrium steady states is
discussed for a simple model of heat flow in a Lorentz gas, previously
described by Spohn and Lebowitz. It is easy to show that for the model
discussed here the two rates are not connected, since the rate of entropy
production is non-zero and positive, while the overall rate of phase space
contraction is zero. This is consistent with conclusions reached by other
workers. Fractal structures appear in the phase space for this model and their
properties are discussed. We conclude with a discussion of the implications of
this and related work for understanding the role of chaotic dynamics and
special initial conditions for an explanation of the Second Law of
Thermodynamics.Comment: 14 pages, 1 figur

### A Transfer Matrix study of the staggered BCSOS model

The phase diagram of the staggered six vertex, or body centered solid on
solid model, is investigated by transfer matrix and finite size scaling
techniques. The phase diagram contains a critical region, bounded by a
Kosterlitz-Thouless line, and a second order line describing a deconstruction
transition. In part of the phase diagram the deconstruction line and the
Kosterlitz-Thouless line approach each other without merging, while the
deconstruction changes its critical behaviour from Ising-like to a different
universality class. Our model has the same type of symmetries as some other
two-dimensional models, such as the fully frustrated XY model, and may be
important for understanding their phase behaviour. The thermal behaviour for
weak staggering is intricate. It may be relevant for the description of
surfaces of ionic crystals of CsCl structure.Comment: 13 pages, RevTex, 1 Postscript file with all figures, to be published
in Phys. Rev.

### Long-time-tail Effects on Lyapunov Exponents of a Random, Two-dimensional Field-driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a
two-dimensional Lorentz gas with randomly placed, non-overlapping hard disk
scatterers placed in a thermostatted electric field, $\vec{E}$. The low density
values of the Lyapunov exponents have been calculated with the use of an
extended Lorentz-Boltzmann equation. In this paper we develop a method to
extend these results to higher density, using the BBGKY hierarchy equations and
extending them to include the additional variables needed for calculation of
Lyapunov exponents. We then consider the effects of correlated collision
sequences, due to the so-called ring events, on the Lyapunov exponents. For
small values of the applied electric field, the ring terms lead to
non-analytic, field dependent, contributions to both the positive and negative
Lyapunov exponents which are of the form ${\tilde{\epsilon}}^{2}
\ln\tilde{\epsilon}$, where $\tilde{\epsilon}$ is a dimensionless parameter
proportional to the strength of the applied field. We show that these
non-analytic terms can be understood as resulting from the change in the
collision frequency from its equilibrium value, due to the presence of the
thermostatted field, and that the collision frequency also contains such
non-analytic terms.Comment: 45 pages, 4 figures, to appear in J. Stat. Phy

### B. B. G. K. Y. Hierarchy Methods for Sums of Lyapunov Exponents for Dilute Gases

We consider a general method for computing the sum of positive Lyapunov
exponents for moderately dense gases. This method is based upon hierarchy
techniques used previously to derive the generalized Boltzmann equation for the
time dependent spatial and velocity distribution functions for such systems. We
extend the variables in the generalized Boltzmann equation to include a new set
of quantities that describe the separation of trajectories in phase space
needed for a calculation of the Lyapunov exponents. The method described here
is especially suitable for calculating the sum of all of the positive Lyapunov
exponents for the system, and may be applied to equilibrium as well as
non-equilibrium situations. For low densities we obtain an extended Boltzmann
equation, from which, under a simplifying approximation, we recover the sum of
positive Lyapunov exponents for hard disk and hard sphere systems, obtained
before by a simpler method. In addition we indicate how to improve these
results by avoiding the simplifying approximation. The restriction to hard
sphere systems in $d$-dimensions is made to keep the somewhat complicated
formalism as clear as possible, but the method can be easily generalized to
apply to gases of particles that interact with strong short range forces.Comment: submitted to CHAOS, special issue, T. Tel. P. Gaspard, and G.
Nicolis, ed

### Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at
low density. The positive Lyapunov exponent may be obtained either by a direct
analysis of the dynamics, or by the use of kinetic theory methods. To leading
orders in the density of scatterers it is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}$, where $A_{0}$ and $B_{0}$ are
known constants and $\tilde{n}$ is the number density of scatterers expressed
in dimensionless units. In this paper, we find that through order
$(\tilde{n}^{2})$, the positive Lyapunov exponent is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}+A_{1}\tilde{n}^{2}\ln\tilde{n}
+B_{1}\tilde{n}^{2}$. Explicit numerical values of the new constants $A_{1}$
and $B_{1}$ are obtained by means of a systematic analysis. This takes into
account, up to $O(\tilde{n}^{2})$, the effects of {\it all\/} possible
trajectories in two versions of the model; in one version overlapping scatterer
configurations are allowed and in the other they are not.Comment: 12 pages, 9 figures, minor changes in this version, to appear in J.
Stat. Phy

### Generalized dynamical entropies in weakly chaotic systems

A large class of technically non-chaotic systems, involving scatterings of
light particles by flat surfaces with sharp boundaries, is nonetheless
characterized by complex random looking motion in phase space. For these
systems one may define a generalized, Tsallis type dynamical entropy that
increases linearly with time. It characterizes a maximal gain of information
about the system that increases as a power of time. However, this entropy
cannot be chosen independently from the choice of coarse graining lengths and
it assigns positive dynamical entropies also to fully integrable systems. By
considering these dependencies in detail one usually will be able to
distinguish weakly chaotic from fully integrable systems.Comment: Submitted to Physica D for the proceedings of the Santa Fe workshop
of November 6-9, 2002 on Anomalous Distributions, Nonlinear Dynamics and
Nonextensivity. 8 pages and two figure